Consider a walk in the square lattice where the
distribution of the fundamental steps is
[[[1, 0], 1/4], [[-1, 0], 1/4], [[0, 1], 1/4], [[0, -1], 1/4]]
Let a(n) be the probability of being at the origin after n steps.
the probability of returning to the origin after n steps .
The linear recurrence operator annihilating a_n:=
2
(n + 1) 2
- -------- + N
2
(n + 2)
The whole thing took, 2.592, seconds of CPU time